A063746 Triangle read by rows giving number of partitions of k (k=0 .. n^2) with Ferrers plot fitting in an n X n box.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 18, 19, 20, 20, 19, 18, 16, 14, 11, 9, 7, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 11, 13, 18, 22, 28, 32, 39, 42, 48, 51, 55, 55, 58, 55, 55, 51, 48, 42, 39, 32, 28

Offset: 0

Author

Wouter Meeussen, Aug 14 2001

Keywords

• nonn
• tabf

Comments

Seems to approximate a Gaussian distribution, the sum of all 1+n^2 terms in a row equals the central binomial coefficients.

a(n,k) is the number of sequences of n 0's and n 1's having major index equal to k (the major index is the sum of the positions of the 1's that are immediately followed by 0's). Equivalently, a(n,k) is the number of Grand Dyck paths of length 2n for which the sum of the positions of the valleys is k. Example: a(3,7)=2 because the only sequences of three 0's and three 1's with major index 7 are 010110 and 110010. The corresponding Grand Dyck paths are obtained by replacing a 0 by a U=(1,1) step and a 1 by a D=(1,-1) step. - Emeric Deutsch, Oct 02 2007

Also, number of n-multisets in [0..n] whose elements sum up to n. - M. F. Hasler, Apr 12 2012

Let P be the poset [n] X [n] ordered by the product order. Let J(P) be the set of all order ideals of P, ordered by inclusion. Then J(P) is a finite sublattice of Young's lattice and T(n,k) is the number of elements in J(P) that have rank k. - Geoffrey Critzer, Mar 26 2020

Examples

			From _M. F. Hasler_, Apr 12 2012: (Start)
The table reads:
n=0: 1  _  (k=0)
n=1: 1 1  _  (k=0..1)
n=2: 1 1 2 1 1  _  (k=0..4)
n=3: 1 1 2 3 3 3 3  2  1  1  _  (k=0..9)
n=4: 1 1 2 3 5 5 7  7  8  7  7  5  5  3  2  1  1  _  (k=0..16)
n=5: 1 1 2 3 5 7 9 11 14 16 18 19 20 20 19 18 16 ...  _  (k=0..25)
etc. (End)
Cycle index of S(3) is (1/6)*(x(1)^3+3*x(1)*x(2)+2*x(3)), so g.f. for 3rd row is (1/6)*((1+x+x^2+x^3)^3+3*(1+x+x^2+x^3)*(1+x^2+x^4+x^6)+2*(1+x^3+x^6+x^9)) = x^9+x^8+2*x^7+3*x^6+3*x^5+3*x^4+3*x^3+2*x^2+x+1.
a(3,7)=2 because the only partitions of 7 with Ferrers plot fitting into a 3 X 3 box are [3,3,1] and [3,2,2].
		

References

• G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 67-69.
• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; exercise 3.2.3.
• A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Links

• Alois P. Heinz, Rows k = 0..31, flattened
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45.
• A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
• A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.

Crossrefs

Cf. A000041, A000070, A002544, A008968, A047971, A047993, A126869, A182738.

Row lengths are given by A002522. - M. F. Hasler, Apr 14 2012

Antidiagonal sums are given by A260894.

Row sums give A000984.

Programs

Maple

for n from 0 to 15 do QBR[n]:=sum(q^i,i=0..n-1) od: for n from 0 to 15 do QFAC[n]:=product(QBR[j],j=1..n) od: qbin:=(n,k)->QFAC[n]/QFAC[k]/QFAC[n-k]: for n from 0 to 7 do P[n]:=sort(expand(simplify(qbin(2*n,n)))) od: for n from 0 to 7 do seq(coeff(P[n],q,j),j=0..n^2) od; # yields sequence in triangular form - Emeric Deutsch, Apr 23 2007
# second Maple program:
b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i-1, k)+
      `if`(i>n, 0, b(n-i, i, k-1))))
    end:
T:= n-> seq(b(k, min(n, k), n), k=0..n^2):
seq(T(n), n=0..8); # Alois P. Heinz, Apr 05 2012

Mathematica

Table[nn=n^2;CoefficientList[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x],{n,0,6}]//Grid (* Geoffrey Critzer, Sep 27 2013 *)
Table[CoefficientList[QBinomial[2n,n,q] // FunctionExpand, q], {n,0,6}] // Flatten (* Peter Luschny, Jul 22 2016 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k - 1]]]];
T[n_] := Table[b[k, Min[n, k], n], {k, 0, n^2}];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

PARI

T(n,k)=polcoeff(prod(i=0,n,sum(j=0,n,x^(j*i*(n^2+n+1)+j),O(x^(k*(n^2+n+1)+n+1)))),k*(n^2+n+1)+n)  /* Based on a more general formula due to R. Gerbicz. M. F. Hasler, Apr 12 2012 */

Formula

Table[T[k, n, n], {n, 0, 9}, {k, 0, n^2}] with T[ ] defined as in A047993.

G.f.: Consider a function; f(n) = 1 + sum(i_1=1, n, sum(i_2=0, i_1, ..., sum(i_n=0, i_(n-1), x^(sum(j=1, n, i_j))*(1+...+x^i_n))...)) Then the GF is f(1)+x^3.f(2)+x^8.f(3)+..., where after x^3 the increase is n^2+1 from f(n). - Jon Perry, Jul 13 2004

G.f. for n-th row is obtained if we set x(i) = 1+x^i+x^(2*i)+...+x^(n*i), i=1, 2, ..., n, in the cycle index Z(S(n);x(1), x(2), ..., x(n)) of the symmetric group S(n) of degree n. - Vladeta Jovovic, Dec 17 2004

G.f. of row n: the q-binomial coefficient [2n,n]. - Emeric Deutsch, Apr 23 2007

T(n,k)=1 for k=0,1,n^2-1,n^2. For all m>n, T(m,n)=T(n,n)=A000041(n), i.e., below the diagonal the columns remain constant, because there cannot be more than n nonzero elements with sum <= n. - M. F. Hasler, Apr 12 2012

T(n,2n) = A128552(n-2). - Geoffrey Critzer, Sep 27 2013

From Alois P. Heinz, Jan 09 2025: (Start)

Sum_{k=0..n} T(n,k) = A000070(n).

Sum_{k=0..n} k * T(n,k) = A182738(n).

Sum_{k=0..n^2} k * T(n,k) = A002544(n-1) for n>=1.

Sum_{k=0..n^2} (-1)^k * T(n,k) = A126869(n). (End)